How to change the system in 4x4 system of first-order equations? [duplicate]












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  • How to change the system in four first order equations?

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Consider the coupled spring-mass system with a frictionless table, two masses $m_1$ and $m_2$, and three springs with spring constants $k_1, k_2$, and $k_3$ respectively. The equation of motion for the system are given by:



$y_1''=-frac{(k_1+k_2)}{m_1}y_1+frac{k_2}{m_1}y_2$



$y_2''=frac{k_2}{m_2}y_1-frac{(k_1+k_2)}{m_2}y_2$



Assume that the masses are $m_1 = 2$, $m_2 = 9/4$, and the spring constants are $k_1=1,k_2=3,k_3=15/4$.



a)use 4x4 system of first order equations to model this system of two second order equations. (hint: $x_1=y_1,x_2=y_2,x_3=y_1',x_4=y_2'$)










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marked as duplicate by LutzL differential-equations
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Dec 5 '18 at 12:35


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.




















    0















    This question already has an answer here:




    • How to change the system in four first order equations?

      2 answers




    Consider the coupled spring-mass system with a frictionless table, two masses $m_1$ and $m_2$, and three springs with spring constants $k_1, k_2$, and $k_3$ respectively. The equation of motion for the system are given by:



    $y_1''=-frac{(k_1+k_2)}{m_1}y_1+frac{k_2}{m_1}y_2$



    $y_2''=frac{k_2}{m_2}y_1-frac{(k_1+k_2)}{m_2}y_2$



    Assume that the masses are $m_1 = 2$, $m_2 = 9/4$, and the spring constants are $k_1=1,k_2=3,k_3=15/4$.



    a)use 4x4 system of first order equations to model this system of two second order equations. (hint: $x_1=y_1,x_2=y_2,x_3=y_1',x_4=y_2'$)










    share|cite|improve this question













    marked as duplicate by LutzL differential-equations
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    Dec 5 '18 at 12:35


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      This question already has an answer here:




      • How to change the system in four first order equations?

        2 answers




      Consider the coupled spring-mass system with a frictionless table, two masses $m_1$ and $m_2$, and three springs with spring constants $k_1, k_2$, and $k_3$ respectively. The equation of motion for the system are given by:



      $y_1''=-frac{(k_1+k_2)}{m_1}y_1+frac{k_2}{m_1}y_2$



      $y_2''=frac{k_2}{m_2}y_1-frac{(k_1+k_2)}{m_2}y_2$



      Assume that the masses are $m_1 = 2$, $m_2 = 9/4$, and the spring constants are $k_1=1,k_2=3,k_3=15/4$.



      a)use 4x4 system of first order equations to model this system of two second order equations. (hint: $x_1=y_1,x_2=y_2,x_3=y_1',x_4=y_2'$)










      share|cite|improve this question














      This question already has an answer here:




      • How to change the system in four first order equations?

        2 answers




      Consider the coupled spring-mass system with a frictionless table, two masses $m_1$ and $m_2$, and three springs with spring constants $k_1, k_2$, and $k_3$ respectively. The equation of motion for the system are given by:



      $y_1''=-frac{(k_1+k_2)}{m_1}y_1+frac{k_2}{m_1}y_2$



      $y_2''=frac{k_2}{m_2}y_1-frac{(k_1+k_2)}{m_2}y_2$



      Assume that the masses are $m_1 = 2$, $m_2 = 9/4$, and the spring constants are $k_1=1,k_2=3,k_3=15/4$.



      a)use 4x4 system of first order equations to model this system of two second order equations. (hint: $x_1=y_1,x_2=y_2,x_3=y_1',x_4=y_2'$)





      This question already has an answer here:




      • How to change the system in four first order equations?

        2 answers








      differential-equations






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      asked Dec 5 '18 at 11:33









      LTYLTY

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      marked as duplicate by LutzL differential-equations
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      Dec 5 '18 at 12:35


      This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.






      marked as duplicate by LutzL differential-equations
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      Dec 5 '18 at 12:35


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          1 Answer
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          0














          Introducing a whole new set of $x_i$ variables is somewhat confusing. Simpler to just introduce two new variables $y_3$ and $y_4$ where $y_3=y_1'$ and $y_4=y_2'$. So you now have:



          $y_1' = y_3$



          $y_2' = y_4$



          $y_3' = y_1'' = dots$



          $y_4'=y_2''= dots$






          share|cite|improve this answer




























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

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            active

            oldest

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            active

            oldest

            votes









            0














            Introducing a whole new set of $x_i$ variables is somewhat confusing. Simpler to just introduce two new variables $y_3$ and $y_4$ where $y_3=y_1'$ and $y_4=y_2'$. So you now have:



            $y_1' = y_3$



            $y_2' = y_4$



            $y_3' = y_1'' = dots$



            $y_4'=y_2''= dots$






            share|cite|improve this answer


























              0














              Introducing a whole new set of $x_i$ variables is somewhat confusing. Simpler to just introduce two new variables $y_3$ and $y_4$ where $y_3=y_1'$ and $y_4=y_2'$. So you now have:



              $y_1' = y_3$



              $y_2' = y_4$



              $y_3' = y_1'' = dots$



              $y_4'=y_2''= dots$






              share|cite|improve this answer
























                0












                0








                0






                Introducing a whole new set of $x_i$ variables is somewhat confusing. Simpler to just introduce two new variables $y_3$ and $y_4$ where $y_3=y_1'$ and $y_4=y_2'$. So you now have:



                $y_1' = y_3$



                $y_2' = y_4$



                $y_3' = y_1'' = dots$



                $y_4'=y_2''= dots$






                share|cite|improve this answer












                Introducing a whole new set of $x_i$ variables is somewhat confusing. Simpler to just introduce two new variables $y_3$ and $y_4$ where $y_3=y_1'$ and $y_4=y_2'$. So you now have:



                $y_1' = y_3$



                $y_2' = y_4$



                $y_3' = y_1'' = dots$



                $y_4'=y_2''= dots$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 5 '18 at 11:57









                gandalf61gandalf61

                7,951625




                7,951625















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